In mathematics, particularly, in asymptotic

convex geometry In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of num ...

, Milman's reverse Brunn–Minkowski inequality is a result due to Vitali Milman that provides a reverse inequality to the famous Brunn–Minkowski inequality for convex bodies in ''n''-

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

R^''n''. Namely, it bounds the volume of the

Minkowski sum In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski ...

of two bodies from above in terms of the volumes of the bodies.

Introduction

Let ''K'' and ''L'' be convex bodies in R^''n''. The Brunn–Minkowski inequality states that :

\mathrm(K+L)^ \geq \mathrm(K)^ + \mathrm(L)^~,

where vol denotes ''n''-dimensional Lebesgue measure and the + on the left-hand side denotes Minkowski addition. In general, no reverse bound is possible, since one can find convex bodies ''K'' and ''L'' of unit volume so that the volume of their Minkowski sum is arbitrarily large. Milman's theorem states that one can replace one of the bodies by its image under a properly chosen volume-preserving

linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...

so that the left-hand side of the Brunn–Minkowski inequality is bounded by a constant multiple of the right-hand side. The result is one of the main structural theorems in the local theory of

Banach spaces In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...

Statement of the inequality

There is a constant ''C'', independent of ''n'', such that for any two centrally symmetric convex bodies ''K'' and ''L'' in R^''n'', there are volume-preserving linear maps ''φ'' and ''ψ'' from R^''n'' to itself such that for any real numbers ''s'', ''t'' > 0 :

\mathrm ( s \, \varphi K + t \, \psi L )^ \leq C \left( s\, \mathrm ( \varphi K )^ + t\, \mathrm ( \psi L )^ \right)~.

One of the maps may be chosen to be the identity.

Notes

References

* * {{DEFAULTSORT:Milman's reverse Brunn-Minkowski inequality Asymptotic geometric analysis Euclidean geometry Geometric inequalities Theorems in measure theory